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Full text of "Treatise On Analysis Vol-Ii"

366        XV    NORMED ALGEBRAS AND SPECTRAL THEORY

have a = ae' (15.8.3(iii)), and on the other hand a = g(e2) = eg(e) = ea, so
that ae eAer, and g =fa. It is clear that eAe' c: eA n Ae'; conversely, if
y e eA n Ae!, then j; = j>e' and y = ey (15.8.3(iii)), so that y = eye' e eAe'.
The image#(I) is a left ideal contained in I', so it is either {0} or I'; likewise,
the kernel ^"^O) is a left ideal contained in I, hence is either {0} or I. If
^-'(O) = I, then g(l) = {0}; if g~l(G) = {0}, then we must have g(l) * {0},
hence^(1) = I' and g is bijective. Finally, if/fl = 0, we have/a(e) = ea = 0; but
a e eAe', so that ea = a, and consequently a = 0.

(ii) The C-algebra eAe is a closed subalgebra of A (15.8.3(iii)). Since we
have seen above that every element of EndA(I) is either zero or invertible, it
follows that EndA(I) is a (possibly noncommutative),/?^, and hence the same
is true for 'eAe. Clearly e is the unit element of eAe, and because A is a
normable algebra (15.1.8) it follows from the Gelfand-Mazur theorem that
eAe = Ce.

(Hi) If I and Y are not isomorphic, we have eAe' = {0} by (i) above,
and in particular eer = 0; hence e and e' are orthogonal (15.8.4). The same is
true of I and I' (15.8.9), and II' = {0}. If I and Y are isomorphic, and if g is an
isomorphism of I onto I7, then every homomorphism of I into I' is of the form
g o u, where u is an endomorphism of I. Hence eAe' is a complex vector space
of dimension 1, by virtue of (i) and (ii) above. Clearly II7 is a left ideal con-
tained in I7; since it contains eAe' ^ {0}, it must be equal to I7.

(iv) Since Ix is the image of I under the homomorphism y^-^-yx of I into
A, it is a left ideal isomorphic to I/I7, where Y is the kernel of the above
homomorphism. But I7 must be equal to either {0} or I, and so y\>yx is
either zero or injective.

(15.8.13)   Suppose that A is separable. Then:

(i) There exists a finite or infinite sequence (Ifc)fceJ of minimal left ideals,
no pair of which are isomorphic, such that every minimal left ideal of A is
isomorphic (as an A-module) to some \k.

(ii) For each index keJ, the closure of the sum of all the minimal left
ideals of A which are isomorphic to \k is a self-adjoint two-sided ideal ak. Every
minimal left ideal of the Hilbert algebra ak is a minimal left ideal of A, isomor-
phic to lk, and the algebra ak contains no closed two-sided ideals other than
{0} and ak.

(iii) Each of the algebras ak is the Hilbert sum of a (finite or infinite)
sequence of minimal left ideals isomorphic to lk. The algebra A is the Hilbert
sum of the sequence (ak)ke], and ahak = {0} whenever h^k.

Start with a decomposition of A as a Hilbert sum of minimal left ideals
l'n (15,8.11). We take It = 1; and define inductively lk+l to be equal to 1^,)n for all